Quantitative Aptitude

Mensuration

2,507 Questions
All Direct Formula Application 1067 Shape Transformations 343 Mensuration Ratio Equations 277 Combined & Hollow Solids 247 Successive Percentage Changes 178 Inscribed & Circumscribed Figures 165 Cost Estimation Problems 100 Liquid Pour & Submergence 67 Unit Conversions 35 Path & Border Calculations 17 Time & Work Integration 7 Scale & Model Proportions 2

Mensuration Questions

Multiple choice

A metal cylinder of radius 8 cm and height 12 cm is melted and made into two solid spheres such that the radius of the first one is half that of the second. What is the ratio of the surface area of the second sphere to the total surface area of the cylinder?

  1. 1 : 3

  2. 1 : 2

  3. 2 : 3

  4. 4 :5

  5. 3 : 4
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The cylinder volume is pi*r^2*h = pi*64*12 = 768*pi. The two spheres have radii r and 2r, so their volumes are (4/3)pi*r^3 and (4/3)*pi(8r^3) = (32/3)*pi*r^3. Summing these gives (36/3)*pi*r^3 = 12*pi*r^3 = 768*pi, so r^3 = 64, r = 4. The second sphere radius is 8. Surface area of second sphere is 4*pi*8^2 = 256*pi. Total surface area of cylinder is 2*pi*r(r+h) = 2*pi*8(8+12) = 320*pi. Ratio is 256/320 = 4/5.

Multiple choice

Mr. Raj has a cylinder and a sphere. The sphere is such that it perfectly fits into the cylinder with no part of it being outside. Then, the ratio of curved surface area of the cylinder to the surface area of the sphere will be

  1. 1 : 1

  2. 1 : 2

  3. 2 : 1

  4. 2 : 3

  5. 3 ; 1
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

If a sphere fits perfectly in a cylinder, the cylinder height h = 2r and radius = r. Surface area of sphere = 4*pi*r^2. Curved surface area of cylinder = 2*pi*r*h = 2*pi*r*(2r) = 4*pi*r^2. Ratio is 1:1.

Multiple choice

A sphere of radius 21 cm is cut into 8 identical parts by 3 cuts (1 cut along each axis). What will be the total surface area (in cm2) of each part?

  1. 844.5

  2. 1732.5

  3. 1039.5

  4. 1115.6
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A sphere cut by three planes along the axes results in 8 octants. Each octant has three flat faces (quarter-circles of radius 21) and one curved surface (one-eighth of the sphere's surface area). Total surface area = 3 * (pi * r^2 / 4) + (4 * pi * r^2 / 8) = (3/4 + 1/2) * pi * r^2 = 1.25 * pi * 21^2 = 1.25 * 441 * 3.14159 = 1731.8, which rounds to 1732.5.

Multiple choice

Two identical hemispheres of maximum possible size are cut from a solid cube of side 14 cm. The bases of the hemispheres are part of the two opposite faces of cube. What is the total volume (in cm3) of the remaining part of the cube?

  1. 1556.33

  2. 898.5

  3. 1467.33

  4. 1306.67
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The volume of the cube is 14^3 = 2744. Two hemispheres of radius 7 cm make one full sphere of volume (4/3) * pi * 7^3 = (4/3) * (22/7) * 343 = 1437.33. Remaining volume = 2744 - 1437.33 = 1306.67.

Multiple choice

Identical cubes of largest possible size are cut from a solid cuboid of size 65 cm × 26 cm × 39 cm. What is the total surface area (in cm2) of all the small cubes taken together?

  1. 30,420

  2. 15,210

  3. 20,280

  4. 16,440
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The side of the largest cube is the HCF of 65, 26, and 39, which is 13 cm. Number of cubes = (65*26*39) / (13*13*13) = 5 * 2 * 3 = 30 cubes. Surface area of one cube = 6 * 13^2 = 6 * 169 = 1014. Total surface area = 30 * 1014 = 30420.

Multiple choice

A right circular cylinder is formed. A = Sum of total surface area and the area of the two bases B = Curved surface area of this cylinder If A : B = 3 : 2 and the volume of cylinder is 4312 cm3, then what is the sum of area (in cm2) of the two bases of this cylinder?

  1. 154

  2. 308

  3. 462

  4. 616
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let r be radius, h be height. A = (2*pi*r^2 + 2*pi*r*h) + 2*pi*r^2 = 4*pi*r^2 + 2*pi*r*h. B = 2*pi*r*h. A/B = (4*pi*r^2 + 2*pi*r*h) / (2*pi*r*h) = 3/2. 2*pi*r^2 / (pi*r*h) + 1 = 1.5 => 2r/h = 0.5 => h = 4r. Volume = pi*r^2*h = pi*r^2*(4r) = 4*pi*r^3 = 4312. r^3 = 4312 / (4 * 22/7) = 1078 * 7 / 22 = 49 * 7 = 343. r = 7. Area of two bases = 2 * pi * r^2 = 2 * 22/7 * 49 = 308.

Multiple choice

A right prism has a square base with side of base 4 cm. The height of the prism is 9 cm. It is cut into three parts of equal height by two planes parallel to its base. What is the ratio of the volumes of the top, middle and bottom parts, respectively?

  1. 1 : 1 : 1

  2. 1 : 7 : 19

  3. 1 : 8 : 20

  4. 1 : 7 : 20
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The prism is cut into three parts of equal height (3 cm each). The base area is 16. Volume of top part = 16 * 3 = 48. Middle part = 16 * 3 = 48. Bottom part = 16 * 3 = 48. The ratio is 1:1:1.

Multiple choice

A hollow cylinder is made up of metal. The difference between the outer and inner curved surface areas of this cylinder is 352 cm2. Height of the cylinder is 28 cm. If the total surface area of this hollow cylinder is 2640 cm2, then what are the inner and outer radii (in cm)?

  1. 4, 6

  2. 10, 12

  3. 8, 10

  4. 6, 8
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let R be the outer radius and r be the inner radius. Difference in curved surface areas: 2*pi*h*(R-r) = 352. 2*(22/7)28(R-r) = 352 => 176*(R-r) = 352 => R-r = 2. Total surface area: 2*pi*(R+r)h + 2*pi(R^2-r^2) = 2640. 2*pi*(R+r)28 + 2*pi(R-r)(R+r) = 2640. 2*pi*(R+r)(28+2) = 2640 => 2(22/7)*(R+r)*30 = 2640 => (R+r) = 2640 * 7 / (44 * 30) = 14. Solving R-r=2 and R+r=14 gives R=8, r=6.

Multiple choice

The total area of a circle and a square is equal to 5450 sq. cm. The diameter of the circle is 70 cm. What is the sum of the circumference of the circle and the perimeter of the square?

  1. 360 cm

  2. 380 cm

  3. 270 cm

  4. Cannot be determined

  5. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Radius = 35 cm. Area of circle = pi * 35^2 = 3850. Area of square = 5450 - 3850 = 1600. Side of square = 40. Circumference = 2 * pi * 35 = 220. Perimeter = 4 * 40 = 160. Sum = 220 + 160 = 380.

Multiple choice

A solid right circular cylinder and a solid hemisphere stand on equal bases and have equal heights. What is the ratio of their total surface areas?

  1. 3 : 2

  2. 3 : 4

  3. 4 : 3

  4. 2 : 3
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let radius be r and height be h. For a cylinder, h = r (since it stands on a hemisphere base). Total surface area of cylinder = 2*pi*r*h + 2*pi*r^2 = 2*pi*r^2 + 2*pi*r^2 = 4*pi*r^2. Total surface area of hemisphere = 3*pi*r^2. The ratio is 4:3.

Multiple choice

The radius of a sphere is 2 cm. If an error of 0.02 cm is made in measuring the radius of the sphere, find the percentage error in measuring its surface area.

  1. ±3%

  2. ±2%

  3. ±1%

  4. ±5%
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The surface area of a sphere is S = 4 * pi * r^2. Differentiating, dS = 8 * pi * r * dr. The relative error is dS/S = (8 * pi * r * dr) / (4 * pi * r^2) = 2 * (dr/r). Given r = 2 and dr = 0.02, the relative error is 2 * (0.02 / 2) = 0.02, which is 2%.